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Quotients of the Booleanization of an inverse semigroup

Ganna Kudryavtseva

Abstract

We introduce $X$-to-join representations of inverse semigroups which are a relaxation of the notion of a cover-to-join representation. We construct the universal $X$-to-join Booleanization of an inverse semigroup $S$ as a weakly meet-preserving quotient of the universal Booleanization ${\mathrm B}(S)$ and show that all such quotients of ${\mathrm B}(S)$ arise via $X$-to-join representaions. As an application, we provide groupoid models for the intermediate boundary quotients of the $C^*$-algebra of a Zappa-Szép product right LCM semigroup by Brownlowe, Ramagge, Robertson and Whittaker.

Quotients of the Booleanization of an inverse semigroup

Abstract

We introduce -to-join representations of inverse semigroups which are a relaxation of the notion of a cover-to-join representation. We construct the universal -to-join Booleanization of an inverse semigroup as a weakly meet-preserving quotient of the universal Booleanization and show that all such quotients of arise via -to-join representaions. As an application, we provide groupoid models for the intermediate boundary quotients of the -algebra of a Zappa-Szép product right LCM semigroup by Brownlowe, Ramagge, Robertson and Whittaker.

Paper Structure

This paper contains 25 sections, 19 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. First definitions and conventions
  4. Representations and proper representations of semilattices
  5. The universal Booleanization of a semilattice
  6. The tight Booleanization of a semilattice
  7. Stone duality for Boolean algebras
  8. Boolean inverse semigroups
  9. Stone groupoids
  10. The universal groupoid of an inverse semigroup
  11. The tight groupoid of an inverse semigroup
  12. Stone duality for Boolean inverse semigroups
  13. $X$-to-join representations of semilattices and inverse semigroups
  14. $X$-to-join representations of semilattices
  15. Connection with $\pi$-tight representations
  16. ...and 10 more sections

Key Result

Proposition 2.7

Suppose that the semilattice $E$ admits the structure of a Boolean algebra. Then a proper representation $\varphi\colon E\to B$ (where $B$ is a Boolean algebra) is tight if and only if it is a morphism between Boolean algebras.

Theorems & Definitions (57)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Proposition 2.7
  • Remark 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 47 more